Question

a check of prices charged for single vision eye glasses at 20 optical centers resulted in the following dollar amounts
190 159 166 171 195 207 141 173 203 161
183 181 198 210 122 170 196 181 195 203
find the 1st quartile and 3rd quartile of the prices
q1 180.25
q3 3605
q1 168
q3 197
q1 180.25
q3 202.35
q1 180.25
q3 190.38

Explanation:

Step1: Sort the data

First, sort the data in ascending order:
122, 141, 159, 161, 166, 170, 171, 173, 181, 181, 181, 183, 190, 195, 195, 196, 198, 203, 203, 207, 210

Step2: Find the position of quartiles

The formula for the position of the \(k\) - th quartile (\(Q_k\)) is \(i=\frac{k(n + 1)}{4}\), where \(n\) is the number of data points. Here \(n = 20\)
For \(Q_1\) (\(k = 1\)): \(i=\frac{1\times(20 + 1)}{4}=5.25\)
For \(Q_3\) (\(k = 3\)): \(i=\frac{3\times(20+ 1)}{4}=15.75\)

Step3: Calculate \(Q_1\)

Since \(i = 5.25\) for \(Q_1\), \(Q_1=x_5+0.25(x_6 - x_5)\)
\(x_5 = 166\), \(x_6=170\)
\(Q_1=166+0.25\times(170 - 166)=166 + 1=167\approx168\)

Step4: Calculate \(Q_3\)

Since \(i = 15.75\) for \(Q_3\), \(Q_3=x_{15}+0.75(x_{16}-x_{15})\)
\(x_{15}=195\), \(x_{16}=196\)
\(Q_3=195+0.75\times(196 - 195)=195 + 0.75=195.75\approx197\)

Answer:

Q1: 168, Q3: 197